Need rigging help for a robotic jaw holder
Various constraints can provide the functionality you are looking for.

For the rigid tubing connected to the main column I would add bones in place to move them. To get them to follow the upper arm movement I would add a Copy Rotation constraint, only enable the axes you want to use and adjust the influence so that they rotate less than the main arm. For the lower piece I would use a StretchTo constraint to keep the end attached to the other tube.

For the flexible tubing going up to the arm, create several small bones to deform the tubing and add a curve to match the rest position of the tubing, then use hooks (see below) to deform the curve and use the curve in a SplineIK constraint to move the bones you positioned for this.





Setting up hooks can be confusing and a little awkward, the step are -


Select the upper arm in pose mode.
Shift select the curve (the outliner can make this easier)
Enter edit mode for the curve.
Select the point in the curve that you want to move with the arm.
Press &#9096; CtrlH->Hook to Selected Object Bone


Now exit edit mode and move the armature, the end of the curve should follow the bone. Note that the viewport lags behind with the splineIK but it will be in the right place when rendering. Repeat for the other end of the curve.